A straight line through the vertex `P` of a triangle `P Q R` intersects the side `Q R` at the points `S` and the cicumcircle of the triangle `P Q R` at the point `Tdot` If `S` is not the center of the circumcircle, then `1/(P S)+1/(S T)<2/(sqrt(Q SxxS R))` `1/(P S)+1/(S T)>2/(sqrt(Q SxxS R))` `1/(P S)+1/(S T)<4/(Q R)` `1/(P S)+1/(S T)>4/(Q R)`

Options

A. `(1)/(PS)+(1)/(ST)lt(2)/(sqrt(QSxxSR))`
B. `(1)/(PS)+(1)/(ST)(2)/(sqrt(QSxxSR))`
C. `(1)/(PS)+(1)/(ST)lt(4)/(QR)`
D. `(1)/(PS)+(1)/(ST)gt(4)/(QR)`

Answer

D

Explanation

Since points P, Q, R and T are concyclic.
`:." "PSxxST=QSxxSR`
Now, `A.MgtG.M,` we have
`(1)/(PS)+(1)/(ST)gt(2)/(sqrt(PSxxST))`
`implies" "(1)/(PS)+(1)/(ST)gt(2)/(sqrt(QSxxSR))" "[becausePSxxST=QSxxSR]" "...(i)`

Again, `A.M.gtG.M.`
`implies" "(QS+SR)/(2)lesqrt(QSxxSR)`
`implies" "(QR)/(2)gesqrt(QSxxSR)implies(2)/(sqrt(QSxxSR))le(4)/(QR)" "...(ii)`
From (i) and (ii), we get `(1)/(PS)+(1)/(ST)gt(4)/(QR)`